Finite Group DatabasebetaFinite Group Databasebeta| order | 540 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C5 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C32×C5×D3 |
| unambiguous semidirect product | C3⋊(C22×C32×C5) |
| unambiguous semidirect product | (C2×C3)⋊(C2×C32×C5) |
| Fitting, socle | C2×C33×C5 ⋊ C2 |
| derived | C3 ⋊ C22×C32×C5 |
| Hall | C33×C5 ⋊ C22 |
| Sylow | C33 ⋊ C22×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32×C5 | D3 |
| commutator, cosocle, hypocenter | C3 | C22×C32×C5 |
| Frattini | C1 | G |
| Fitting, socle | C2×C33×C5 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C32×C5 | |
152 conjugacy classes of subgroups.
| transitive degree | μt(G) = 180 |
270 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C32×D3×Dic5, C32×#1201, C32×#1203, … |
| maximal normal subgroup of | C32×D3×Dic5, C32×#1201, C32×#1203, … |
| maximal quotient of | C32×C5×Dic6, C32×C4×C5×D3, C32×C5×D12, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(540, 113);